# Polyharmonic functions for finite graphs and Markov chains

## TL;DR

This paper introduces and analyzes polyharmonic functions on finite graphs and Markov chains, solving boundary value problems and comparing with infinite tree cases, expanding the understanding of discrete potential theory.

## Contribution

It defines polyharmonic functions for finite graphs and Markov chains, and solves the Riquier boundary value problem in this discrete setting.

## Key findings

- Unique solutions to boundary problems are established.
- Polyharmonic functions are characterized in finite graph and Markov chain contexts.
- Connections to infinite tree boundary theory are explored.

## Abstract

On a finite graph with a chosen partition of the vertex set into interior and boundary vertices, a $\lambda$-polyharmonic function is a complex function $f$ on the vertex set which satisfies $(\lambda \cdot I - P)^n f(x) = 0$ at each interior vertex. Here, $P$ may be the normalised adjaceny matrix, but more generally, we consider the transition matrix $P$ of an arbitrary Markov chain to which the (oriented) graph structure is adapted. After describing these `global' polyharmonic functions, we turn to solving the Riquier problem, where $n$ boundary functions are preassigned and a corresponding `tower' of $n$ successive Dirichlet type problems are solved. The resulting unique solution will be polyharmonic only at those points which have distance at least $n$ from the boundary. Finally, we compare these results with those concerning infinite trees with the end boundary, as studied by Cohen, Colonnna, Gowrisankaran and Singman, and more recently, by Picardello and Woess.